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Schrödinger operator with magnetic field in domain with corners

Virginie Bonnaillie Noël (2005)

Journées Équations aux dérivées partielles

We present here a simplified version of results obtained with F. Alouges, M. Dauge, B. Helffer and G. Vial (cf [4, 7, 9]). We analyze the Schrödinger operator with magnetic field in an infinite sector. This study allows to determine accurate approximation of the low-lying eigenpairs of the Schrödinger operator in domains with corners. We complete this analysis with numerical experiments.

Semiclassical distribution of eigenvalues for elliptic operators with Hölder continuous coefficients, part i: non-critical case

Lech Zieliński (2004)

Colloquium Mathematicae

We consider a version of the Weyl formula describing the asymptotic behaviour of the counting function of eigenvalues in the semiclassical approximation for self-adjoint elliptic differential operators under weak regularity hypotheses. Our aim is to treat Hölder continuous coefficients and to investigate the case of critical energy values as well.

Sharp spectral asymptotics and Weyl formula for elliptic operators with Non-smooth Coefficients-Part 2

Lech Zielinski (2002)

Colloquium Mathematicae

We describe the asymptotic distribution of eigenvalues of self-adjoint elliptic differential operators, assuming that the first-order derivatives of the coefficients are Lipschitz continuous. We consider the asymptotic formula of Hörmander's type for the spectral function of pseudodifferential operators obtained via a regularization procedure of non-smooth coefficients.

Sharp trace asymptotics for a class of 2 D -magnetic operators

Horia D. Cornean, Søren Fournais, Rupert L. Frank, Bernard Helffer (2013)

Annales de l’institut Fourier

In this paper we prove a two-term asymptotic formula for the spectral counting function for a 2 D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2 D Fermi gas submitted to a constant external magnetic field.The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure...

Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that...

Some remarks on the Weyl asymptotics by the approximate spectral projection method

Ernesto Buzano (2000)

Bollettino dell'Unione Matematica Italiana

In questo lavoro studiamo il resto relativo della formula asintotica per gli autovalori di un operatore differenziale in R n , ottenuta mediante il metodo delle proiezioni spettrali approssimate ([3], Theorem 6.2). In un primo tempo diamo un controesempio di un operatore di Schrödinger con potenziale a crescita algebrica, per il quale il resto non è limitato. Quindi specifichiamo alcune condizioni addizionali da imporre all'operatore in modo da avere un resto infinitesimo.

Spectral analysis in a thin domain with periodically oscillating characteristics

Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

Spectral analysis in a thin domain with periodically oscillating characteristics

Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). ...

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